Twisted symmetric trilayer graphene. II. Projected Hartree-Fock study

The Hamiltonian of the magic-angle twisted symmetric trilayer graphene (TSTG) can be decomposed into a TBG-like flat band and high-velocity Dirac fermion Hamiltonian. We use Hartree-Fock mean field approach to study projected Coulomb interacting TSTG developed in C\u{a}lug\u{a}ru et al. [Phys. Rev. B 103, 195411 (2021)] at integer fillings $\nu=-3, -2, -1$ $0$ measured from charge neutrality. phase diagram with $w_0/w_1$, ratio $AA$ $AB$ interlayer hoppings, displacement field, which introduces an potential $U$ hybridizes bands bands. At small $U$, we find ground states all $\nu$ are same phases as tensor products semimetal filling TBG insulator states, spin-valley polarized $\nu=-3$, fully (partially) intervalley coherent $\nu=-2,0$ ($\nu=-1$) An exception is $\nu=-3$ $w_0/w_1 \gtrsim 0.7$, possibly become metal competing orders due transfers between strong where bandwidths exceed interactions, enter or zero valley polarization coherence. Lastly, intermediate coherence may arise for $\nu=-2,-1,0$. Our results provide simple picture electron interactions systems, reveal connection states.